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The real number system encompasses both rational and irrational numbers, each with unique characteristics. Rational numbers can be expressed as fractions with integer numerators and denominators, including integers, finite decimals, and repeating decimals. Irrational numbers, such as π and √2, have non-repeating, non-terminating decimal expansions. This system is fundamental to mathematics, with properties like closure, commutative, associative, and distributive that govern operations.

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## Classification of Numbers

### Rational Numbers

Rational numbers can be expressed as a fraction with integer numerators and non-zero integer denominators

### Irrational Numbers

Irrational numbers have decimal expansions that neither terminate nor repeat

### Real Numbers

Real numbers encompass both rational and irrational numbers on the number line

## Subsets of Real Numbers

### Natural Numbers

Natural numbers are positive integers beginning with 1

### Whole Numbers

Whole numbers include all natural numbers plus zero

### Integers

Integers comprise positive and negative whole numbers, including zero

## Properties of Real Numbers

### Closure Property

The closure property states that the sum or product of any two real numbers is a real number

### Commutative Property

The commutative property asserts that the order in which two numbers are added or multiplied does not affect the result

### Associative Property

The associative property allows for the grouping of numbers to be altered in addition or multiplication without changing the outcome

### Distributive Property

The distributive property links multiplication with addition and subtraction, facilitating the simplification of expressions

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